Lesson 2Circular Grid
Learning Goal
Let’s dilate figures on circular grids.
Learning Targets
I can apply dilations to figures on a circular grid when the center of dilation is the center of the grid.
Lesson Terms
- center of dilation
- dilation
- scale factor
Warm Up: Notice and Wonder: Concentric Circles
Problem 1
What do you notice? What do you wonder?
Activity 1: A Droplet on the Surface
Problem 1
The larger circle
Draw four points on the smaller circle using the Point on Object tool.
Draw the rays from
through each of those four points. Select the Ray tool, then point , and then the second point. Mark the intersection points of the rays and circle d by selecting the Intersect tool and clicking on the point of intersection.
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The larger Circle
Draw four points on the smaller circle (not inside the circle!), and label them
, , , and . Draw the rays from
through each of those four points. Label the points where the rays meet the larger circle
, , , and .
Problem 2
Complete the table. In the row labeled
Problem 3
The center of dilation is point
Activity 2: Quadrilateral on a Circular Grid
Problem 1
Here is a polygon
Dilate each vertex of polygon
using as the center of dilation and a scale factor of 2. Draw segments between the dilated points to create a new polygon.
What are some things you notice about the new polygon?
Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
Dilate each vertex of polygon
using as the center of dilation and a scale factor of . What do you notice about this new polygon?
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Here is a polygon
Dilate each vertex of polygon
using as the center of dilation and a scale factor of 2. Label the image of as , and label the images of the remaining three vertices as , , and . Draw segments between the dilated points to create polygon
.
What are some things you notice about the new polygon?
Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
Dilate each vertex of polygon
using as the center of dilation and a scale factor of . Label the image of as , the image of as , the image of as and the image of as . What do you notice about polygon
?
Are you ready for more?
Problem 1
Suppose
is twice as long . is five units longer than . The point
is on . and intersect.
Activity 3: A Quadrilateral and Concentric Circles
Problem 1
Dilate polygon
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Dilate polygon
Lesson Summary
A circular grid like this one can be helpful for performing dilations.
The radius of the smallest circle is one unit, and the radius of each successive circle is one unit more than the previous one.
To perform a dilation, we need a center of dilation, a scale factor, and a point to dilate. In the picture,
Since the circles on the grid are the same distance apart, segment